D₂₆₉₀₄ #72290243

order	53808
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₉₀₄

Splittings

Fitting	C₃×C₈×C₁₉×C₅₉ ⋊ C₂
Hall	C₃×C₁₉×C₅₉ ⋊ D₈
Hall	C₁₉×C₅₉ ⋊ D₂₄
Hall	C₃×C₅₉ ⋊ D₁₅₂
Sylow	C₅₉ ⋊ D₄₅₆
Hall	C₃×C₁₉ ⋊ D₄₇₂
Sylow	C₁₉ ⋊ D₁₄₁₆
Sylow	C₃ ⋊ D₈₉₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₄₅₂
commutator, cosocle	C₃×C₄×C₁₉×C₅₉	C₂²
Frattini	C₄	C₂×D₃₃₆₃
Fitting	C₃×C₈×C₁₉×C₅₉	C₂
socle	C₂×C₃×C₁₉×C₅₉	D₄
hypercenter	C₈	D₃₃₆₃
hypocenter	C₃×C₁₉×C₅₉	D₈
LCS	G ▷ C₃×C₄×C₁₉×C₅₉ ▷ C₂×C₃×C₁₉×C₅₉ ▷ C₃×C₁₉×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 26904

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(2,8)(3,7)(4,6)(10,11)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2^4,3,19,59|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^18,g^a=g^58

Conjugacy classes & character table

13455 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 19, 59]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26904 #72290243